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Massless chiral excitations can arise from the interactions between a fermionand an axion string, propagating along the string and allowing it tosuperconduct. The properties of these excitations, or zero modes, dictate howthe string interacts with light and can thus have important phenomenologicalconsequences. In this paper, we add a nowhere-vanishing Dirac mass for thefermion in the usual model of axion electrodynamics. We find that the zeromodes exhibit an interesting phase structure in which they delocalize from thestring's core as the mass increases, up until a critical value past which theydisappear. We study this structure from an analytic perspective, with explicitnumerical solutions, and via anomaly inflow arguments. Finally, we derive thetwo-dimensional effective theory of the zero mode and its interactions with thefour-dimensional gauge field and show how this effective theory breaks down asthe zero modes delocalize.

Zero Modes of Massive Fermions Delocalize from Axion Strings